Integrand size = 39, antiderivative size = 366 \[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2}}{d f (5+2 m)}-\frac {2^{\frac {1}{2}+m} a (2 c (C+2 C m)+d (C (3-2 m)+A (5+2 m))) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2}-m,-\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (1+\sin (e+f x))^{\frac {1}{2}-m} (a+a \sin (e+f x))^{-1+m} \sqrt {c+d \sin (e+f x)}}{d f (5+2 m) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2^{\frac {5}{2}+m} C (d m-c (1+m)) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2}-m,-\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)}}{d f (5+2 m) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \] Output:
-2*C*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(3/2)/d/f/(5+2*m)-2^(1 /2+m)*a*(2*c*(2*C*m+C)+d*(C*(3-2*m)+A*(5+2*m)))*AppellF1(1/2,-1/2,1/2-m,3/ 2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*cos(f*x+e)*(1+sin(f*x+e))^(1/ 2-m)*(a+a*sin(f*x+e))^(-1+m)*(c+d*sin(f*x+e))^(1/2)/d/f/(5+2*m)/((c+d*sin( f*x+e))/(c+d))^(1/2)-2^(5/2+m)*C*(d*m-c*(1+m))*AppellF1(1/2,-1/2,-1/2-m,3/ 2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin(f*x+e))*cos(f*x+e)*(1+sin(f*x+e))^(-1 /2-m)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)/d/f/(5+2*m)/((c+d*sin(f*x+ e))/(c+d))^(1/2)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx \] Input:
Integrate[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + C*Sin[e + f *x]^2),x]
Output:
Integrate[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + C*Sin[e + f *x]^2), x]
Time = 1.18 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3042, 3525, 27, 3042, 3466, 3042, 3267, 157, 27, 156, 155}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a \sin (e+f x)+a)^m \left (A+C \sin ^2(e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a)^m \left (A+C \sin (e+f x)^2\right ) \sqrt {c+d \sin (e+f x)}dx\) |
\(\Big \downarrow \) 3525 |
\(\displaystyle \frac {2 \int \frac {1}{2} (\sin (e+f x) a+a)^m \sqrt {c+d \sin (e+f x)} (a (A d (2 m+5)+C (3 d+2 c m))+2 a C (d m-c (m+1)) \sin (e+f x))dx}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int (\sin (e+f x) a+a)^m \sqrt {c+d \sin (e+f x)} (a (A d (2 m+5)+C (3 d+2 c m))+2 a C (d m-c (m+1)) \sin (e+f x))dx}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (\sin (e+f x) a+a)^m \sqrt {c+d \sin (e+f x)} (a (A d (2 m+5)+C (3 d+2 c m))+2 a C (d m-c (m+1)) \sin (e+f x))dx}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 3466 |
\(\displaystyle \frac {a (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \int (\sin (e+f x) a+a)^m \sqrt {c+d \sin (e+f x)}dx+2 C (d m-c (m+1)) \int (\sin (e+f x) a+a)^{m+1} \sqrt {c+d \sin (e+f x)}dx}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \int (\sin (e+f x) a+a)^m \sqrt {c+d \sin (e+f x)}dx+2 C (d m-c (m+1)) \int (\sin (e+f x) a+a)^{m+1} \sqrt {c+d \sin (e+f x)}dx}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 3267 |
\(\displaystyle \frac {\frac {a^3 \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 C (d m-c (m+1)) \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {a-a \sin (e+f x)}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 157 |
\(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \int \frac {\sqrt {2} (\sin (e+f x) a+a)^{m-\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}+\frac {\sqrt {2} a^2 C (d m-c (m+1)) \sqrt {1-\sin (e+f x)} \cos (e+f x) \int \frac {\sqrt {2} (\sin (e+f x) a+a)^{m+\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 C (d m-c (m+1)) \sqrt {1-\sin (e+f x)} \cos (e+f x) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}} \sqrt {c+d \sin (e+f x)}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) \sqrt {c+d \sin (e+f x)} \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}} \sqrt {\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {2 a^2 C (d m-c (m+1)) \sqrt {1-\sin (e+f x)} \cos (e+f x) \sqrt {c+d \sin (e+f x)} \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}} \sqrt {\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle \frac {\frac {\sqrt {2} a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) (A d (2 m+5)+2 c (2 C m+C)+C d (3-2 m)) (a \sin (e+f x)+a)^m \sqrt {c+d \sin (e+f x)} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},-\frac {1}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+1) (a-a \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {2 \sqrt {2} a C (d m-c (m+1)) \sqrt {1-\sin (e+f x)} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt {c+d \sin (e+f x)} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},-\frac {1}{2},m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+3) (a-a \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}}{a d (2 m+5)}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{3/2}}{d f (2 m+5)}\) |
Input:
Int[(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]]*(A + C*Sin[e + f*x]^2) ,x]
Output:
(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2))/(d*f *(5 + 2*m)) + ((Sqrt[2]*a^2*(C*d*(3 - 2*m) + A*d*(5 + 2*m) + 2*c*(C + 2*C* m))*AppellF1[1/2 + m, 1/2, -1/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*(a - a*Sin[e + f*x])*Sqrt [(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*a*C*(d*m - c*(1 + m))*AppellF 1[3/2 + m, 1/2, -1/2, 5/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x ]))/(c - d))]*Cos[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(f*(3 + 2*m)*(a - a*Sin[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c - d)]))/(a*d*(5 + 2*m))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n ]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & !GtQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x] && !Si mplerQ[e + f*x, a + b*x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d* x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m , n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A*b - a*B)/b Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x ] + Simp[B/b Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {c +d \sin \left (f x +e \right )}\, \left (A +C \sin \left (f x +e \right )^{2}\right )d x\]
Input:
int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x)
Output:
int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x, algorithm="fricas")
Output:
integral(-(C*cos(f*x + e)^2 - A - C)*sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + C \sin ^{2}{\left (e + f x \right )}\right ) \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \] Input:
integrate((a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**(1/2)*(A+C*sin(f*x+e)**2), x)
Output:
Integral((a*(sin(e + f*x) + 1))**m*(A + C*sin(e + f*x)**2)*sqrt(c + d*sin( e + f*x)), x)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x, algorithm="maxima")
Output:
integrate((C*sin(f*x + e)^2 + A)*sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x, algorithm="giac")
Output:
integrate((C*sin(f*x + e)^2 + A)*sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)
Timed out. \[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int \left (C\,{\sin \left (e+f\,x\right )}^2+A\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^(1/ 2),x)
Output:
int((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m*(c + d*sin(e + f*x))^(1/ 2), x)
\[ \int (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \left (A+C \sin ^2(e+f x)\right ) \, dx=\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \left (a +a \sin \left (f x +e \right )\right )^{m} \sin \left (f x +e \right )^{2}d x \right ) c +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \left (a +a \sin \left (f x +e \right )\right )^{m}d x \right ) a \] Input:
int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2)*(A+C*sin(f*x+e)^2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)*(sin(e + f*x)*a + a)**m*sin(e + f*x)**2,x)*c + int(sqrt(sin(e + f*x)*d + c)*(sin(e + f*x)*a + a)**m,x)*a